Strong Convergence of Parallel Iterative Algorithm with Mean Errors for Two Finite Families of Ćirić Quasi-Contractive Operators

and Applied Analysis 3 where {αn} be a real sequence in [0, 1]. If ∑∞ n 1 αn ∞, then {xn}n 0 converges strongly to the unique fixed point of T . In 2006, Rafiq 7 considered a class of mappings satisfying the following condition: ∥ ∥Tx − Ty∥ ≤ hmax { ∥ ∥x − y∥, ‖x − Tx‖ ∥ ∥y − Ty∥ 2 , ∥ ∥x − Ty∥,∥y − Tx∥ } , CR 0 < h < 1. This class of mappings is a subclass of mappings satisfying the following condition: ∥ ∥Tx − Ty∥ ≤ hmax∥x − y∥, ‖x − Tx‖,∥y − Ty∥,∥x − Ty∥,∥y − Tx∥, CQ 0 < h < 1. The class of mappings satisfying CQ was introduced and investigated by Ćirić 8 in 1974 and a mapping satisfying is commonly called Ćirić quasi-contraction. Rafiq 7 proved the following result. TheoremR see 7 . LetC be a nonempty closed convex subset of a normed space E. Let T : C → C be an operator satisfying the condition CR . For given x0 ∈ C, let {xn} be generated by the algorithm xn 1 αnxn βnTxn γnun, n ≥ 0, 1.6 where {αn}, {βn}, and {γn} be three real sequences in [0, 1] satisfying αn βn γn 1 for all n ≥ 1, {un} is a bounded sequences in C. If ∑∞ n 1 βn ∞ and γn o αn , then {xn}n 0 converges strongly to the unique fixed point of T . In 2007, Gu 9 proved the following theorem. Theorem G see 9 . Let C be a nonempty closed convex subset of a normed space E. Let {Ti}i 1 : C → C beN operators satisfying the condition CR with F ∩i 1F Ti / ∅ (the set of common fixed points of {Ti}i 1). Let {αn}, {βn}, and {γn} be three real sequences in [0, 1] satisfying αn βn γn 1 for all n ≥ 1, {un} a bounded sequences in C satisfying the following conditions: i ∑∞ n 1 βn ∞; ii ∑∞ n 1 γn < ∞ or γn o βn . Suppose further that x0 ∈ C is any given point and {xn} is generated by the algorithm xn 1 αnxn βnTnxn γnun, n ≥ 0, 1.7 where Tn Tn mod N . Then {xn} converges strongly to a common fixed point of {Ti}i 1. Remark 1.4. It should be pointed out that Theorem G extends Theorem R from a Ćirić quasicontractive operator to a finite family of Ćirić quasi-contractive operators. 4 Abstract and Applied Analysis Inspired andmotivated by the facts said above, we introduced a new two-step parallel iterative algorithm with mean errors for two finite family of operators {Si}i 1 and {Tj}j 1 as follows: xn 1 ( 1 − αn − γn ) xn αn m ∑ i 1 λiSiyn γnun, n ≥ 1, yn ( 1 − βn − δn ) xn βn k ∑ j 1 μiTjxn δnvn, n ≥ 1, 1.8 where {λi}i 1, {μj}j 1 are two finite sequences of positive number such that ∑m i 1 λi 1 and ∑k j 1 μj 1, {αn}, {βn}, {γn} and {δn} are four real sequences in 0, 1 satisfying αn γn ≤ 1 and βn δn ≤ 1 for all n ≥ 1, {un} and {vn} are two bounded sequences in C and x0 is a given point. Especially, if {αn}, {γn} are two sequences in 0, 1 satisfying αn γn ≤ 1 for all n ≥ 1, {λi}i 1 ⊂ 0, 1 satisfying λ1 λ2 · · · λm 1, {un} is a bounded sequence in C and x0 is a given point in C, then the sequence {xn} defined by xn 1 ( 1 − αn − γn ) xn αn m ∑ i 1 λiSixn γnun, n ≥ 1 1.9 is called the one-step parallel iterative algorithm with mean errors for a finite family of operators {Si}i 1. The purpose of this paper is to study the convergence of two-steps parallel iterative algorithm with mean errors defined by 1.8 to a common fixed point for two finite family of Ćirić quasi-contractive operators in normed spaces. The results presented in this paper generalized and extend the corresponding results of Berinde 5 , Gu 9 , Rafiq 7 , Rhoades 10 , and Zamfirescu 3 . Even in the case of βn δn 0 or γn δn 0 for all n ≥ 1 or m k 1 are also new. In order to prove the main results of this paper, we need the following Lemma. Lemma 1.5 see 11 . Suppose that {an}, {bn}, and {cn} are three nonnegative real sequences satisfying the following condition: an 1 ≤ 1 − tn an bn cn, ∀n ≥ n0, 1.10 where n0 is some nonnegative integer, tn ∈ 0, 1 , ∑∞ n 0 tn ∞, bn o tn and ∑∞ n 0 cn < ∞. Then limn→∞an 0. 2. Main Results We are now in a position to prove our main results in this paper. Theorem 2.1. Let C be a nonempty closed convex subset of a normed space E. Let {Si}i 1 : C → C be m operators satisfying the condition CR and {Tj}j 1 : C → C be k operators satisfying the Abstract and Applied Analysis 5 condition CR with F ⋂m i 1 F Si ∩ ⋂k j 1 F Tj / ∅, where F Si and F Tj are the set of fixed points of Si and Tj in C, respectively. Let {αn}, {βn}, {γn}, and {δn} be four real sequences in [0, 1] satisfying αn γn ≤ 1 and βn δn ≤ 1 for all n ≥ 1, {λi}i 1, {μj}j 1 two finite sequences of positive number such that ∑m i 1 λi 1 and ∑k j 1 μj 1, {un} and {vn} two bounded sequences in C satisfying the following conditions: i ∑∞ n 1 αn ∞; ii limn→∞δn 0; iii ∑∞ n 1 γn < ∞ or γn o αn . Suppose further that x0 ∈ C is any given point and {xn} is an iteration sequence with mane errors defined by 1.8 , then {xn} converges strongly to a common fixed point of {Si}i 1 and {Tj}j 1. Proof. Since {Si}i 1 : C → C is m Ćirić operator satisfying the condition CR , hence there exists 0 < hi < 1 i ∈ I {1, 2, . . . , m} such that ∥Six − Siy ∥ ≤ hi max { ∥x − y∥, ‖x − Six‖ ∥y − Siy ∥ 2 , ∥x − Siy ∥, ∥y − Six ∥ } . 2.1and Applied Analysis 5 condition CR with F ⋂m i 1 F Si ∩ ⋂k j 1 F Tj / ∅, where F Si and F Tj are the set of fixed points of Si and Tj in C, respectively. Let {αn}, {βn}, {γn}, and {δn} be four real sequences in [0, 1] satisfying αn γn ≤ 1 and βn δn ≤ 1 for all n ≥ 1, {λi}i 1, {μj}j 1 two finite sequences of positive number such that ∑m i 1 λi 1 and ∑k j 1 μj 1, {un} and {vn} two bounded sequences in C satisfying the following conditions: i ∑∞ n 1 αn ∞; ii limn→∞δn 0; iii ∑∞ n 1 γn < ∞ or γn o αn . Suppose further that x0 ∈ C is any given point and {xn} is an iteration sequence with mane errors defined by 1.8 , then {xn} converges strongly to a common fixed point of {Si}i 1 and {Tj}j 1. Proof. Since {Si}i 1 : C → C is m Ćirić operator satisfying the condition CR , hence there exists 0 < hi < 1 i ∈ I {1, 2, . . . , m} such that ∥Six − Siy ∥ ≤ hi max { ∥x − y∥, ‖x − Six‖ ∥y − Siy ∥ 2 , ∥x − Siy ∥, ∥y − Six ∥ } . 2.1 For each fixed i ∈ I {1, 2, . . . , m}. Denote h max{h1, h2, . . . , hm}, then 0 < h < 1 and ∥Six − Siy ∥ ≤ hmax { ∥x − y∥, ‖x − Six‖ ∥y − Siy ∥ 2 , ∥x − Siy ∥, ∥y − Six ∥ } 2.2 hold for each fixed i ∈ I {1, 2, . . . , m}. If from 2.2 we have ∥Six − Siy ∥ ≤ h 2 ‖x − Six‖ ∥y − Siy ∥, 2.3 then ∥Six − Siy ∥ ≤ h 2 ‖x − Six‖ ∥y − Siy ∥] ≤ h 2 ‖x − Six‖ ∥y − x∥ ‖x − Six‖ ∥Six − Siy ∥. 2.4 Hence ( 1 − h 2 )∥∥Six − Siy ∥∥ ≤ h 2 ∥x − y∥ h‖x − Six‖, 2.5 which yields using the fact that 0 < h < 1 ∥Six − Siy ∥ ≤ h/2 1 − h/2 ∥x − y∥ h 1 − h/2‖x − Six‖. 2.6 6 Abstract and Applied Analysis Also, from 2.2 , if ∥ ∥Six − Siy ∥ ∥ ≤ hmax∥x − Siy ∥ ∥, ∥ ∥y − Six ∥ ∥} 2.7 holds, then a ‖Six − Siy‖ ≤ h‖x − Siy‖, which implies ‖Six − Siy‖ ≤ h‖x − Six‖ h‖Six − Siy‖ and hence, as h < 1, ∥ ∥Six − Siy ∥ ∥ ≤ h 1 − h‖x − Six‖, 2.8 or b ‖Six − Siy‖ ≤ h‖y − Six‖, which implies ∥Six − Siy ∥∥ ≤ h∥y − x∥ h‖x − Six‖. 2.9 Thus, if 2.7 holds, then from 2.8 and 2.9 we have ∥Six − Siy ∥ ≤ h∥y − x∥ h 1 − h‖x − Six‖. 2.10